It is well known that such a running coupling leads to unphysical singularities like the Landau pole [1]}, which renders perturbation theory useless for low-energy processes in QCD. A resolution of this problem was proposed a long time ago [2]}, when the RG method was unified with the requirement of analyticity. This led to the concept of an analytic running coupling \(\bar{\alpha }(Q^2)\)  that is free of unphysical singularities. Subsequently, other methods have been developed through the use of dispersion relations that reflect the causality principle [3]}, [4]}, [5]}, [6]}, and successfully applied in QCD. The analytic approach preserves the ultraviolet behavior in the ultraviolet region, which ensures asymptotic freedom but leads to essential modifications of the behaviour in the infrared region.
There are no unphysical singularities in the low energy region.
An important feature of this approach is the presence of a universal value of the analytic coupling \(\bar{\alpha }(Q^2 = 0) = 4 \pi /\beta _0\) , where \(\beta _0\)  is the coefficient of the \(\beta \) -function at one loop.
Several arguments for the universality of the analytic running coupling have been previously given in in the literature, from various points of view [3]}, [4]}, [5]}, [6]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}.
Another relevant property of the analytic approach is that it allows us to define the causal running coupling \(\bar{\alpha }(Q^2)\)  both in the space-like as well as in the time-like domains, in a consistent way. These features enable meaningful calculations of hadronic processes in QCD, like the inelastic lepton-hadron scattering and the electron-positron annihilation into hadrons as described, for example, in the reviews [18]}, [19]} and the references cited therein.
